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Question Number 51215 by gunawan last updated on 25/Dec/18

$${\int }_0^{\pi }{e}^{(1+i)x}dx=...$$

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Dec/18

$$a=1+i$$$${\int }_0^{\pi }{e}^{ax}dx$$$$=\frac{1}{a}\mid e^{ax}{\mid }_0^{\pi }$$$$=\frac{1}{a}(e^{a\pi }-1)$$$$=\frac{1}{1+i}(e^{(1+i)\pi }-1)$$$$=\frac{1-i}{2}[e^{\pi }{e}^{i\pi }-1][e^{i\pi }=cos\pi +isin\pi =-1]$$$$=\frac{1-i}{2}(-e^{\pi }-1)$$$$=\frac{-1+i}{2}(e^{\pi }+1)$$$$$$

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